Efficiency-based Comparison and Design of Spatial Networks

Many networked systems are embedded into space [1]. One of their hallmarks is the existence of a cost associated to the edge’s length which, in turns, affects the topology of the connectivity pattern. Given a fixed amount of total resources (i.e. total link length Ltot), there are multiple ways of building a spatial network satisfying such constraint. Therefore, a criterion to compare the design of different spatial networks for a given node layout and a fixed link length is needed. In this contribution, we address this issue by proposing: (1) a comprehensive measure of the efficiency of spatial networks; (2) an algorithm to build the (almost) most efficient network possible for a given total cost. Up to now, a number of different measures accounting for complementary properties related with the efficiency of spatial networks have been introduced [2]. We propose a combined estimator that accounts for both the quality of the normal functioning of the network at a global scale and its robustness against failures at local scale:

Specifically, the efficiency of a network depends on the distance, in the two-dimensional space (EL, EG), between its coordinates and the complete graph [by definition, located at (1,1)](see Figure 1). In order to assess whether the design of a network is better than another, a direct comparison between systems with different node layout and total link length could be not only unfair, but also not informative. On the contrary, the availability of a model to generate the best constrained network enables an indirect comparison. At this aim, we have devised a model that creates links by optimizing the use of resources in a computationally feasible way for an arbitrary node layout.